To address this question, from here onwards we assume that \((M,g)\)  has been isometrically embedded in some closed Riemannian manifold \((N,g)\) . This can be done for instance by constructing the double of the manifold \(M\)  as explained in [1]} and extending the metric \(g\)  smoothly across the boundary \(\partial M\) . The issue with this extension is that it might create `short cuts' in the sense that there can be a curve in \(N\) , connecting some points of \(M\) , which is shorter than any curve entirely contained in \(M\) . Therefore we always have
\(d_{M}(p,q)\ge d_{N}(p,q), \quad \hbox{ for all } p,q \in M,\) 
